The topology of Stein fillable manifolds in high dimensions, II

We continue our study of contact structures on manifolds of dimension at least five using surgery-theoretic methods. Particular applications include the existence of “maximal” almost contact manifolds with respect to the Stein cobordism relation as well as the existence of weakly fillable contact structures on the product M S 2 . We also study the connection between Stein fillability and connected sums: we give examples of almost contact manifolds for which the connected sum is Stein fillable, while the components are not. Concerning obstructions to Stein fillability, we show for all k > 1 that there are almost contact structures on the .8k1/–sphere which are not Stein fillable. This implies the same result for all highly connected .8k1/–manifolds which admit almost contact structures. The proofs rely on a new number-theoretic result about Bernoulli numbers. 32E10; 57R17, 57R65